MATH 844: HOMEWORK 1, DUE FEB 2
1. Suppose that (a, b) is a rational point on Reichardts curve x4 17 = 2y 2 .
(a) Show that a = A/C, b = B/C 2 , where A, B, C Z, gcd(A, B ) = gcd(A, C ) =
gcd(B, C ) = 1.
(b) Rewriting the equation as (5A2 + 17C 2 )2 (4B )
MATH 844: HOMEWORK 6, DUE MAR 9.
6. Call a smooth projective curve C over Fq maximal if |C (Fq )| = q + 1 + 2g q
(g being its genus).
(a) Show that there are no maximal curves of positive genus over F2 .
(b) Show that the Hermitian curve xq+1 + y q+1 + z
MATH 844: HOMEWORK 5, DUE MAR 2.
5. (a) Let g = x3 y + y 3 z + z 3 x Fq [x, y, z ]. Let V be the projective curve given
by g = 0. Find the zeta function of V for q = 2 and q = 3.
(b) Let f = xq+1 y q y Fq2 [x, y ] (q a prime power). Let C be the projectiv
MATH 844: HOMEWORK 3, DUE FEB 16
3. Consider the problem of nding triples of positive integers (a, b, c) with no
common factor such that the triangle ABC with sides a, b, c has the following property:
() The median from A, the angle bisector from B, and t
MATH 844: HOMEWORK 2, DUE FEB 9
2. (a) Prove that all the roots e1 , e2 , e3 of 4x3 g2 x g3 are real and distinct if
3
2
and only if g2 and g3 are real and := g2 27g3 > 0.
(b) Suppose the conditions in (a) are met, and order the ei so that e2 > e3 > e1 .
MATH 844: HOMEWORK 10, DUE APR 13.
10. Every year, the London Sunday Telegraph has a New Years Quiz. In 1995,
two of the questions were the following:
(a) Solve the equation A3 /B 3 + C 3 /D3 = 6, where A, B, C, D are all positive
whole numbers below 100.
MATH 844: HOMEWORK 7, DUE MAR 23.
7. Let y 2 = f (x) dene an elliptic curve E over Q, where f (x) Z[x] is a cubic
polynomial.
(a) If E has good reduction at p, show that
|E (Fp )| = 1 + p +
(
f (x)
)
p
where () is the Legendre symbol and the sum is over a
MATH 844: HOMEWORK 9, DUE APR 6.
9. Let E be the elliptic curve y 2 = x3 + 16.
(a) Show that this equation denes an elliptic curve over Fp (p prime) if and only
if p 5.
(b) Calculate a5 and a7 where ap = p + 1 |E (Fp )|.
(c) Which kind of singularity does
MATH 844: HOMEWORK 11, DUE APR 20.
11. The integer equation a4 + ma2 b2 + b4 = c2 () (a, b) = 1, a, b > 0 was studied
by Fermat and Euler. A solution is called trivial if either ab = 0 or a = b = 1.
(a) Let E be the elliptic curve over Q given by y 2 = x3
MATH 844: HOMEWORK 8, DUE MAR 30.
8. Let E be an elliptic curve over Fq .
(a) Show that E (Fq ) Z/mZ/mn for some integers m, n 1 with gcd(m, q ) =
=
1.
(b) With the notation of (a), show that q 1 (mod m).
(c) Suppose that q is a prime 5 and that E is supe
MATH 844: HOMEWORK 4, DUE FEB 23
4. Let E be the elliptic curve y 2 + y = x3 x, with group law identity (0 : 1 : 0).
Let Km denote the extension of Q generated by the coordinates of all the points P
in E [m] := cfw_P E (Q) | mP = identity.
(a) Show that K